SIAM ReviewPub Date : 2022-02-01DOI: 10.1137/20m1342902
C. Kelley
{"title":"Newton's Method in Mixed Precision","authors":"C. Kelley","doi":"10.1137/20m1342902","DOIUrl":"https://doi.org/10.1137/20m1342902","url":null,"abstract":". We investigate the use of reduced precision arithmetic to solve the linear equation for the Newton step. If one 3 neglects the backward error in the linear solve, then well-known convergence theory implies that using single precision in the 4 linear solve has very little negative e(cid:11)ect on the nonlinear convergence rate. 5 However, if one considers the e(cid:11)ects of backward error, then the usual textbook estimates are very pessimistic and even the 6 state-of-the-art estimates using probabilistic rounding analysis do not fully conform to experiments. We report on experiments 7 with a speci(cid:12)c example. We store and factor Jacobians in double, single, and half precision. In the single precision case we 8 observe that the convergence rates for the nonlinear iteration do not degrade as the dimension increases and that the nonlinear 9 iteration statistics are essentially identical to the double precision computation. In half precision we see that the nonlinear 10 convergence rates, while poor, do not degrade as the dimension increases. 11 Audience. This paper is intended for students who have completed or are taking an entry-level graduate course in 12 numerical analysis and for faculty who teach numerical analysis. The important ideas in the paper are O notation, (cid:13)oating 13 point precision, backward error in linear solvers, and Newton’s method.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"59 1","pages":"191-211"},"PeriodicalIF":10.2,"publicationDate":"2022-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86218883","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2021-12-14DOI: 10.1137/20M1360888
Joseph D. Johnson, Adam M. Redlich, D. Abrams
{"title":"A Mathematical Model for the Origin of Name Brands and Generics","authors":"Joseph D. Johnson, Adam M. Redlich, D. Abrams","doi":"10.1137/20M1360888","DOIUrl":"https://doi.org/10.1137/20M1360888","url":null,"abstract":"Firms in the U.S. spend over 200 billion dollars each year advertising their products to consumers, around one percent of the country's gross domestic product. It is of great interest to understand how that aggregate expenditure affects prices, market efficiency, and overall welfare. Here, we present a mathematical model for the dynamics of competition through advertising and find a surprising prediction: when advertising is relatively cheap compared to the maximum benefit advertising offers, rational firms split into two groups, one with significantly less advertising (a\"generic\"group) and one with significantly more advertising (a\"name brand\"group). Our model predicts that this segmentation will also be reflected in price distributions; we use large consumer data sets to test this prediction and find good qualitative agreement.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"32 4 1","pages":"625-639"},"PeriodicalIF":10.2,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77405376","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2021-07-24DOI: 10.1137/21M1436087
W. Clark, Mario W. Gomes, A. Rodriguez-Gonzalez, L. Stein, S. Strogatz
{"title":"Surprises in a Classic Boundary-Layer Problem","authors":"W. Clark, Mario W. Gomes, A. Rodriguez-Gonzalez, L. Stein, S. Strogatz","doi":"10.1137/21M1436087","DOIUrl":"https://doi.org/10.1137/21M1436087","url":null,"abstract":"We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"8 1","pages":"291-315"},"PeriodicalIF":10.2,"publicationDate":"2021-07-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90626121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2021-02-24DOI: 10.1137/21m1401243
S. Brunton, M. Budišić, E. Kaiser, J. Kutz
{"title":"Modern Koopman Theory for Dynamical Systems","authors":"S. Brunton, M. Budišić, E. Kaiser, J. Kutz","doi":"10.1137/21m1401243","DOIUrl":"https://doi.org/10.1137/21m1401243","url":null,"abstract":"The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinite-dimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"48 1","pages":"229-340"},"PeriodicalIF":10.2,"publicationDate":"2021-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75247032","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2021-01-23DOI: 10.1137/20m1389522
L. Trefethen
{"title":"Exactness of quadrature formulas","authors":"L. Trefethen","doi":"10.1137/20m1389522","DOIUrl":"https://doi.org/10.1137/20m1389522","url":null,"abstract":"The standard design principle for quadrature formulas is that they should be exact for integrands of a given class, such as polynomials of a fixed degree. We show how this principle fails to predict the actual behavior in four cases: Newton–Cotes, Clenshaw–Curtis, Gauss–Legendre, and Gauss–Hermite quadrature. Three further examples are mentioned more briefly.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"28 1","pages":"132-150"},"PeriodicalIF":10.2,"publicationDate":"2021-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74731560","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM ReviewPub Date : 2021-01-07DOI: 10.1137/21m1389778
R. Glowinski, Yongcun Song, Xiaoming Yuan, Hangrui Yue
{"title":"Bilinear Optimal Control of an Advection-Reaction-Diffusion System","authors":"R. Glowinski, Yongcun Song, Xiaoming Yuan, Hangrui Yue","doi":"10.1137/21m1389778","DOIUrl":"https://doi.org/10.1137/21m1389778","url":null,"abstract":"We consider the bilinear optimal control of an advection-reaction-diffusion system, where the control arises as the velocity field in the advection term. Such a problem is generally challenging from both theoretical analysis and algorithmic design perspectives mainly because the state variable depends nonlinearly on the control variable and an additional divergence-free constraint on the control is coupled together with the state equation. Mathematically, the proof of the existence of optimal solutions is delicate, and up to now, only some results are known for a few special cases where additional restrictions are imposed on the space dimension and the regularity of the control. We prove the existence of optimal controls and derive the first-order optimality conditions in general settings without any extra assumption. Computationally, the well-known conjugate gradient (CG) method can be applied conceptually. However, due to the additional divergence-free constraint on the control variable and the nonlinear relation between the state and control variables, it is challenging to compute the gradient and the optimal stepsize at each CG iteration, and thus nontrivial to implement the CG method. To address these issues, we advocate a fast inner preconditioned CG method to ensure the divergence-free constraint and an efficient inexactness strategy to determine an appropriate stepsize. An easily implementable nested CG method is thus proposed for solving such a complicated problem. For the numerical discretization, we combine finite difference methods for the time discretization and finite element methods for the space discretization. Efficiency of the proposed nested CG method is promisingly validated by the results of some preliminary numerical experiments.","PeriodicalId":49525,"journal":{"name":"SIAM Review","volume":"69 1","pages":"392-421"},"PeriodicalIF":10.2,"publicationDate":"2021-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86452920","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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